## Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution.(English)Zbl 0960.90079

In this paper, some notions of well-posedness are studied for parametric variational inequalities $$VI(x)$$ and for optimization problems with variational inequalities constraints $$OPVIC$$. The problem $$VI(x)$$ is defined by the pair $$(A(x,u),K)$$, where $$A(x,.)$$ is an operator from $$E$$ to $$E^{\ast }$$ and $$K\subset E$$ is a nonempty closed convex set. The $$OPVIC$$ is intended as minimizing the function $$f(x,u)$$ over the set $$\{(x,u)\in X\times K\mid u\in T(x)\}$$, where $$T(x)\subset E$$ is the solution set of $$VI(x)$$. In both cases the variational inequalities considered are supposed to be uniquely solvable.
The first notion studied is the parametrically strongly well-posedness of the family $$VI(x)$$, which is proven to be a generalization of the similar definition given by T. Zolezzi [Nonl. Anal., Theory Meth. Appl. 25, 437-453 (1995; Zbl 0841.49005)] for the case of parametric optimization problems. The authors give a characterization of the parametrically strongly well-posedness of $$VI(x)$$ for finite dimensional $$E$$ and a sufficient condition for the case $$A(u)$$ does not depend on $$x$$. For the latter case it is also given another characterization of the introduced concept in terms of the diameter of an $$\epsilon$$-solution set defined in a former paper. This last characterization can be extended only as a necessary condition to the general case $$A(x,u)$$.
In a second section the authors introduce the concept of approximating sequences for $$OPVIC$$, which generalizes the same notion used in a former paper by the second author for bilevel programming problems. The notions of generalized and strongly well-posedness of $$OPVIC$$ are defined and sufficient conditions are provided. Both concepts are also characterized in case of finite dimensional $$E$$. Finally, an application of the introduced concepts to an exact penalty method is shortly presented.

### MSC:

 90C30 Nonlinear programming 90C31 Sensitivity, stability, parametric optimization 58E35 Variational inequalities (global problems) in infinite-dimensional spaces

Zbl 0841.49005
Full Text: